\(\mathcal{YiHsin}\;\mathcal{Lu}\)

1 Testing

Please load the attached data by load test.RData Analyze the attached data by the linear model on the response variable (\(Y\)) with two predictor variables (\(X1, X2\)). Also use the appropriate statistical method to check whether the corresponding coefficient of \(X1\) (\(\beta_1\)) is greater than 2.

【solution】

  • Since the purpose is test whether \(\beta_1\) greater than 2 or not. If our testing result reject \(H_0\), then we are finished.

  • \(\Large{H_0:\beta_1\leq2}\)

  • \(\Large{H_1:\beta_1>2}\)

Base on the data

\(y_i = \beta_0+\beta_1x_{i1}+\beta_2x_{i2}\quad\Rightarrow\quad\beta_{i1} = \cfrac{y_i-\hat{\beta_0}-\hat{\beta_2}x_{i2}}{x_{i1}}\quad\), use \(\bar{\beta_1}\) to estimate \(\beta_1\), and use t-test to check \(H_0\).

beta1 = (Y-beta_h[1]-beta_h[3]*x2)/x1
t.test(beta1, mu = 2, alternative = 'greater')
## 
##  One Sample t-test
## 
## data:  beta1
## t = 29.263, df = 499, p-value < 2.2e-16
## alternative hypothesis: true mean is greater than 2
## 95 percent confidence interval:
##  2.287514      Inf
## sample estimates:
## mean of x 
##  2.304671
qt(p = 0.05, df = 499, lower.tail = F)
## [1] 1.647913

Since \(t_{\beta_1} = 29.263>1.647913=t_{0.05,499}\)

Reject \(H_0\), accept \(H_1\), then the corresponding coefficient of \(X1\) (\(\beta_1\)) is greater than 2

Linear Model t-test

  • \(\large{y_i = \beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\epsilon_i\;,\;\forall \;i=1,\cdots,500}\)
summary(linear_model)
## 
## Call:
## lm(formula = y ~ ., data = data_model)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.2287 -1.3926 -0.5757  0.7399 10.0556 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.39836    1.16459   2.059   0.0400 *  
## x1           2.30468    0.09954  23.153   <2e-16 ***
## x2          -0.20463    0.09924  -2.062   0.0397 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.929 on 497 degrees of freedom
## Multiple R-squared:  0.5192, Adjusted R-squared:  0.5172 
## F-statistic: 268.3 on 2 and 497 DF,  p-value: < 2.2e-16

3D plot

fig

2 Summary Report

Moment Generating Function

【Definition】


Let \(\mathbb{X} = (X_1,X_2,\cdots,X_n)^T\) be random vector and \(\mathbb{t} = (t_1,t_2,\cdots,t_n)^T\in\mathbb{R}^n\), MGF (Moment Generating Function) is defined by: \[ M_\mathbb{X}(t) = E(e^{t^T\mathbb{X}}) \] for all \(t\) for which the expectation exists (finite).

【Theorem1】


Let \(M_\mathbb{X}(t)=M_\mathbb{Y}(t)\;,\;\;\forall\;t\in(-t_0,t_0)^n\), then: \[ F_\mathbb{X}(z)=F_\mathbb{Y}(z)\;,\;\forall z\in\mathbb{R}^n \] where \(F_\mathbb{X}(z)\) and \(F_\mathbb{Y}(z)\): joint cdfs of \(\mathbb{X}\) and \(\mathbb{Y}\)

【Theorem2】


Let \(X_1,X_2,\cdots,X_m\in\mathbb{R}^n\) are independent , and \(\mathbb{X} = X_1+X_2+\cdots+X_m\). Then: \[ M_\mathbb{X}(t) = \prod_{i=1}^m M_{X_i}(t) \]

【Theorem3】


Let * \(\mathbb{X}\): random vector in \(\mathbb{R}^n\) * \(A\): \(m\times n\) real matrix * \(b\in\mathbb{R}^m\) * \(\mathbb{Y} = A\mathbb{X}+b\) * \(t\in\mathbb{R}^m\). Then: \[ M_\mathbb{Y}(t) = e^{\mathbb{t}^Tb}M_\mathbb{X}(A^Tt) \]

Multivariate Normal Distribution

【Theorem4】


If \(\mathbb{X}=(X_1,X_2,\cdots,X_n)^T \sim N(\mu,\Sigma)\), the joint pdf given by: \[ f_\mathbb{X}(x) = \cfrac{1}{\sqrt{(2\pi)^n\;det\;\Sigma}}e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)}\;,\qquad x\in\mathbb{R}^n \]